The frequency of the current and the amount of power passed in an electrical winding during electromagnetic transfer, be it for power transmission like in a transformer, or for the generation of heat in a workpiece, as with induction heating, are major factors which determine the size, dimensions and internal structure of an electrical winding, or coil.
While the cylindrical shape is most representative of an electrical winding, more often than not it is a mere approximation and the art is replete with coils which have ampere-turns established in a multilayer fashion of different current cross sections and overall geometry, especially where conductor insulation and cooling affect the general dimensions. Nevertheless, the superposition of ampere-turns about a common axis as if the overall shape of the winding were cylindrical, has the merit of providing a good account of the electrical characteristics and the efficiency of an electrical winding of any design. In this regard, ampere-turn dimensions, laterally and radially of the axis, are essential from a point of view of the total magnetomotive force, of field intensity, and current losses. Representative of the prior art is an Article by R. M. Baker in AIEE Transactions, Volume 26 Part II, March 1957, pp. 31-40, entitled "Design and Calculation of Induction-Heating Coils". R. M. Baker in the article, in the context of induction heating applications, considers the flux in the air gap between coil and workpoiece, the work flux which is effective on the workpiece itself, and also the flux in the copper of the coil due to magnetic field penetration. In this respect, the author distinguishes two types of effective depth of current penetration .delta.: .delta..sub.c in the coil copper and .delta..sub.w in the workpiece. It is realized, indeed, that skin effect causes the induced current to flow in a restricted manner more or less close to the surface depending upon the field intensity and frequency. Therefore, the geometric disposition of the copper, the air gap and the workpiece are essential consideration to measure coil effectiveness. This appears from the design calculations in the Baker article, involving the effective depth of current penetration .delta., and current density in the copper, as well as the external factors affecting the flux. The following formula is given: ##EQU1## in centimeters, where .rho.=electrical resistivity (ohm-cm); .mu.=relative effective magnetic permeability (.mu.=1 in air or other non-magnetic materials like copper, brass, aluminum; .mu. is between 10 and 100 for iron and steel); f=frequency in cycles/sec.
Skin effect in the coil is the reverse of the same effect in the workpiece. Magnetic field intensity and current density are both maximum on the inner radius of the coil turns and drop off exponentially along the radius outwardly. It is .delta..sub.c calculated for the depth of current penetration in copper which determines this current distribution. Likewise, .delta..sub.w around the periphery of the workpiece is known. It follows that coil and workpiece resistances are known. They determine the associated losses in the copper and effective heat generation RI.sup.2 in the workpiece.
The problem of losses becomes particularly acute in induction heating where high current densities are encountered. An approach to cope with this problem has been to use multiple layers of thin strap conductors to reduce power losses in the winding. However, this is at the expense of water cooling which cannot be easily accommodated on, or between, such thin layers of copper. This is in contrast to present cooling practice consisting of using rectangular copper cross-sections allowing a round, or rectangular, axial "hole" in the copper through which cooling water is forced to flow. Therefore, it is desirable to minimize power losses in a coil having rectangular-shaped copper cross-sections arranged for a central cooling passage.
In keeping with ampere turns having a definite thickness to accommodate inner copper cooling, the present invention takes advantage of the conclusions reached by Ketalin Gallyas in a thesis delivered at the University of Toronto, Canada, entitled "Current Density and Power Loss Distribution in Sheet Windings". In this paper, the author has developed a theory for the minimization of losses in multilayer windings. The model used for such theory consists of sheets of copper arranged in a multilayer fashion to form the "coil", and an optimum thickness for the layers is calculated. The optimum layer thickness b.sub.opt for a coil with q layers is given by the formulae: EQU b.sub.opt =1.3.delta./q
when q.gtoreq.2, and EQU b.sub.opt =1.57.delta.,
q when q=1.
As a result of such optimization, Gallyas has shown that the ratio of loss P.sub.q in a winding of q layers, as opposed to the loss P.sub.1 in a single layer winding is: ##EQU2##
The conclusions of Gallyas in her thesis have been based on a winding supporting the same current in each layer, while all layers have been given the same b.sub.opt thickness. The results obtained by Gallyas are as shown in Table I herebelow:
TABLE I ______________________________________ Number of Loss Ratio for Fixed Current layers (q) and Thickness (P.sub.q /P.sub.1) ______________________________________ 1 1.000 2 .856 3 .699 4 .605 5 .541 6 .494 7 .457 8 .428 ______________________________________